Standard Error of the Mean Calculator

Understanding the standard error of the mean — concept and principle

The standard error of the mean (SEM) measures how precisely a sample mean estimates the true population mean. The standard deviation describes how much individual values vary; the SEM describes how much the mean itself would vary if you repeated the study with new samples of the same size. It is used to build confidence intervals, to draw error bars and to plan how large a sample must be. Enter the sample standard deviation and the sample size.

With 36 people the mean’s wobble shrinks to 12 ÷ 6 = 2, a 95% margin of ±3.92.

How it is calculated

SEM = s ÷ √n, where s is the sample standard deviation and n the number of observations.
95% margin of error ≈ 1.96 × SEM. Adding and subtracting it from the sample mean gives an approximate 95% confidence interval.

The SEM has the same unit as the data.

The principle

Averaging cancels part of the random variation. If observations are independent with standard deviation σ, the variance of their sum is nσ², so the variance of the mean is nσ² ÷ n² = σ² ÷ n and its standard deviation is σ ÷ √n. By the central limit theorem the sample mean is close to normally distributed for moderately large n, even when the data are not, and about 95% of a normal distribution lies within 1.96 standard deviations of the centre. That is where the 1.96 factor comes from.

Worked example

Inputs: s = 12, n = 36.
√36 = 6.
SEM = 12 ÷ 6 = 2.
95% margin = 1.96 × 2 = 3.92.
For illustration, a sample mean of 100 would give the interval 96.08 to 103.92.

Each point is the calculator re-run with only “Standard deviation” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Standard deviation	SEM
3	0.5
6	1
9	1.5
12	2
15	2.5
18	3
24	4

How to read the result

A small SEM means the sample mean is a precise estimate. Because of the square root, precision improves slowly: to halve the SEM you need four times as many observations (here 144 instead of 36). In error bars, ±1 SEM shows only about a 68% range, so always state whether bars show SD, SEM or a 95% confidence interval. Non-overlapping ±1 SEM bars do not by themselves prove a significant difference.

Common mistakes

Reporting the SEM in place of the SD to make data look less variable.
Using the population formula when the data are clustered or not independent.
Using 1.96 with very small samples instead of the t distribution value.

Keep in mind: The 1.96 factor is a large-sample normal approximation. With small samples use the t critical value with n − 1 degrees of freedom; for n = 36 it is about 2.03, giving a margin of about 4.06. The formula also assumes a simple random sample; survey designs with clustering or weighting need adjusted standard errors.

Frequently asked questions

What is the difference between standard deviation and standard error?

The SD describes spread among individual observations; the SE describes uncertainty in an estimate such as the mean, and it shrinks as n grows.

Can I get the SD back from the SEM?

Yes, SD = SEM × √n.

Is there a standard error for other statistics?

Yes. Proportions, regression coefficients and differences of means all have their own standard error formulas.

Sources: Altman, D. G. & Bland, J. M. (2005). Standard deviations and standard errors. BMJ, 331, 903. · NIST/SEMATECH e-Handbook of Statistical Methods.

Checked 8 Nov 2026. Standard formulas, computed in your browser with double-precision numbers.

Your numbers